Conway group Co1

In the area of modern algebra known as group theory, the Conway group Co1 is a sporadic simple group of order

   2²¹ · 3⁹ · 5⁴ · 7² · 11 · 13 · 23

= 4157776806543360000

≈ 4×10¹⁸.

History and properties

Co1 is one of the 26 sporadic groups and was discovered by John Horton Conway in 1968. It is the largest of the three sporadic Conway groups and can be obtained as the quotient of Co0 (group of automorphisms of the Leech lattice Λ that fix the origin) by its center, which consists of the scalar matrices ±1. It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice II25,1. Some rather cryptic comments in Witt's collected works suggest that he found the Leech lattice and possibly the order of its automorphism group in unpublished work in 1940.

The outer automorphism group is trivial and the Schur multiplier has order 2.
Involutions

Co₀ has 4 conjugacy classes of involutions; these collapse to 2 in Co₁, but there are 4-elements in Co₀ that correspond to a third class of involutions in Co₁.

An image of a dodecad has a centralizer of type 2¹¹:M₁₂:2, which is contained in a maximal subgroup of type 2¹¹:M₂₄.

An image of an octad or 16-set has a centralizer of the form 2¹⁺⁸.O₈⁺(2), a maximal subgroup.

Representations

The smallest faithful permutation representation of Co₁ is on the 98280 pairs {v,–v} of norm 4 vectors.

There is a matrix representation of dimension 24 over the field {\displaystyle \mathbb {F} _{2}} \)

The centralizer of an involution of type 2B in the monster group is of the form 2¹⁺²⁴Co₁.

The Dynkin diagram of the even Lorentzian unimodular lattice II_1,25 is isometric to the (affine) Leech lattice Λ, so the group of diagram automorphisms is split extension Λ,Co₀ of affine isometries of the Leech lattice.

Maximal subgroups

Wilson (1983) found the 22 conjugacy classes of maximal subgroups of Co₁, though there were some errors in this list, corrected by Wilson (1988).

• Co₂
• 3.Suz:2 The lift to Aut(Λ) = Co₀ fixes a complex structure or changes it to the complex conjugate structure. Also, top of Suzuki chain.
• 2¹¹:M₂₄ Image of monomial subgroup from Aut(Λ), that subgroup stabilizing the standard frame of 48 vectors of form (±8,0²³) .
• Co₃
• 2¹⁺⁸.O₈⁺(2) centralizer of involution class 2A (image of octad from Aut(Λ))
• Fi₂₁:S₃ ≈ U₆(2):S₃ The lift to Aut(Λ) is the symmetry group of a coplanar hexagon of 6 type 2 points.
• (A₄ × G₂(4)):2 in Suzuki chain.
• 2²⁺¹²:(A₈ × S₃)
• 2⁴⁺¹².(S₃ × 3.S₆)
• 3².U₄(3).D₈
• 3⁶:2.M₁₂ (holomorph of ternary Golay code)
• (A₅ × J₂):2 in Suzuki chain
• 3¹⁺⁴:2.PSp₄(3).2
• (A₆ × U₃(3)).2 in Suzuki chain
• 3³⁺⁴:2.(S₄ × S₄)
• A₉ × S₃ in Suzuki chain
• (A₇ × L₂(7)):2 in Suzuki chain
• (D₁₀ × (A₅ × A₅).2).2
• 5¹⁺²:GL₂(5)
• 5³:(4 × A₅).2
• 7²:(3 × 2.S₄)
• 5²:2A₅

References

Conway, John Horton (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups", Proceedings of the National Academy of Sciences of the United States of America, 61 (2): 398–400, doi:10.1073/pnas.61.2.398, MR 0237634, PMC 225171, PMID 16591697
Brauer, R.; Sah, Chih-han, eds. (1969), Theory of finite groups: A symposium, W. A. Benjamin, Inc., New York-Amsterdam, MR 0240186
Conway, John Horton (1969), "A group of order 8,315,553,613,086,720,000", The Bulletin of the London Mathematical Society, 1: 79–88, doi:10.1112/blms/1.1.79, ISSN 0024-6093, MR 0248216
Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham (eds.), Finite simple groups, Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 215–247, ISBN 978-0-12-563850-0, MR 0338152 Reprinted in Conway & Sloane (1999, 267-298)
Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, MR 0920369
Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, 21, Mathematical Association of America, ISBN 978-0-88385-023-7, MR 0749038
Conway, John Horton; Parker, Richard A.; Norton, Simon P.; Curtis, R. T.; Wilson, Robert A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, MR 0827219
Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03516-0, ISBN 978-3-540-62778-4, MR 1707296
Wilson, Robert A. (1983), "The maximal subgroups of Conway's group Co₁", Journal of Algebra, 85 (1): 144–165, doi:10.1016/0021-8693(83)90122-9, ISSN 0021-8693, MR 0723071
Wilson, Robert A. (1988), "On the 3-local subgroups of Conway's group Co₁", Journal of Algebra, 113 (1): 261–262, doi:10.1016/0021-8693(88)90192-5, ISSN 0021-8693, MR 0928064
Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012

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